Algebra Examples

\frac{4 x + 1}{x - 2} + \frac{2}{x} = \frac{- 4}{x^{2} - 2 x}

$\frac{4 x + 1}{x - 2} + \frac{2}{x} = \frac{- 4}{x^{2} - 2 x}$

Step 1

Subtract

\frac{- 4}{x^{2} - 2 x}

$\frac{- 4}{x^{2} - 2 x}$ from both sides of the equation.

\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x^{2} - 2 x} = 0

$\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x^{2} - 2 x} = 0$

Step 2

Simplify

\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x^{2} - 2 x}

$\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x^{2} - 2 x}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Factor

x

$x$ out of

x^{2} - 2 x

$x^{2} - 2 x$ .

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Step 2.1.1.1

Factor

x

$x$ out of

x^{2}

$x^{2}$ .

\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x \cdot x - 2 x} = 0

$\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x \cdot x - 2 x} = 0$

Step 2.1.1.2

Factor

x

$x$ out of

- 2 x

$- 2 x$ .

\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x \cdot x + x \cdot - 2} = 0

$\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x \cdot x + x \cdot - 2} = 0$

Step 2.1.1.3

Factor

x

$x$ out of

x \cdot x + x \cdot - 2

$x \cdot x + x \cdot - 2$ .

\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x (x - 2)} = 0

$\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x (x - 2)} = 0$

\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x (x - 2)} = 0

$\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x (x - 2)} = 0$

Step 2.1.2

Move the negative in front of the fraction.

\frac{4 x + 1}{x - 2} + \frac{2}{x} - - \frac{4}{x (x - 2)} = 0

$\frac{4 x + 1}{x - 2} + \frac{2}{x} - - \frac{4}{x (x - 2)} = 0$

Step 2.1.3

Multiply

- - \frac{4}{x (x - 2)}

$- - \frac{4}{x (x - 2)}$ .

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Step 2.1.3.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{4 x + 1}{x - 2} + \frac{2}{x} + 1 \frac{4}{x (x - 2)} = 0

$\frac{4 x + 1}{x - 2} + \frac{2}{x} + 1 \frac{4}{x (x - 2)} = 0$

Step 2.1.3.2

Multiply

\frac{4}{x (x - 2)}

$\frac{4}{x (x - 2)}$ by

1

$1$ .

\frac{4 x + 1}{x - 2} + \frac{2}{x} + \frac{4}{x (x - 2)} = 0

$\frac{4 x + 1}{x - 2} + \frac{2}{x} + \frac{4}{x (x - 2)} = 0$

\frac{4 x + 1}{x - 2} + \frac{2}{x} + \frac{4}{x (x - 2)} = 0

$\frac{4 x + 1}{x - 2} + \frac{2}{x} + \frac{4}{x (x - 2)} = 0$

\frac{4 x + 1}{x - 2} + \frac{2}{x} + \frac{4}{x (x - 2)} = 0

$\frac{4 x + 1}{x - 2} + \frac{2}{x} + \frac{4}{x (x - 2)} = 0$

Step 2.2

To write

\frac{4 x + 1}{x - 2}

$\frac{4 x + 1}{x - 2}$ as a fraction with a common denominator, multiply by

\frac{x}{x}

$\frac{x}{x}$ .

\frac{4 x + 1}{x - 2} \cdot \frac{x}{x} + \frac{2}{x} + \frac{4}{x (x - 2)} = 0

$\frac{4 x + 1}{x - 2} \cdot \frac{x}{x} + \frac{2}{x} + \frac{4}{x (x - 2)} = 0$

Step 2.3

To write

\frac{2}{x}

$\frac{2}{x}$ as a fraction with a common denominator, multiply by

\frac{x - 2}{x - 2}

$\frac{x - 2}{x - 2}$ .

\frac{4 x + 1}{x - 2} \cdot \frac{x}{x} + \frac{2}{x} \cdot \frac{x - 2}{x - 2} + \frac{4}{x (x - 2)} = 0

$\frac{4 x + 1}{x - 2} \cdot \frac{x}{x} + \frac{2}{x} \cdot \frac{x - 2}{x - 2} + \frac{4}{x (x - 2)} = 0$

Step 2.4

Write each expression with a common denominator of

(x - 2) x

$(x - 2) x$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 2.4.1

Multiply

\frac{4 x + 1}{x - 2}

$\frac{4 x + 1}{x - 2}$ by

\frac{x}{x}

$\frac{x}{x}$ .

\frac{(4 x + 1) x}{(x - 2) x} + \frac{2}{x} \cdot \frac{x - 2}{x - 2} + \frac{4}{x (x - 2)} = 0

$\frac{(4 x + 1) x}{(x - 2) x} + \frac{2}{x} \cdot \frac{x - 2}{x - 2} + \frac{4}{x (x - 2)} = 0$

Step 2.4.2

Multiply

\frac{2}{x}

$\frac{2}{x}$ by

\frac{x - 2}{x - 2}

$\frac{x - 2}{x - 2}$ .

\frac{(4 x + 1) x}{(x - 2) x} + \frac{2 (x - 2)}{x (x - 2)} + \frac{4}{x (x - 2)} = 0

$\frac{(4 x + 1) x}{(x - 2) x} + \frac{2 (x - 2)}{x (x - 2)} + \frac{4}{x (x - 2)} = 0$

Step 2.4.3

Reorder the factors of

(x - 2) x

$(x - 2) x$ .

\frac{(4 x + 1) x}{x (x - 2)} + \frac{2 (x - 2)}{x (x - 2)} + \frac{4}{x (x - 2)} = 0

$\frac{(4 x + 1) x}{x (x - 2)} + \frac{2 (x - 2)}{x (x - 2)} + \frac{4}{x (x - 2)} = 0$

\frac{(4 x + 1) x}{x (x - 2)} + \frac{2 (x - 2)}{x (x - 2)} + \frac{4}{x (x - 2)} = 0

$\frac{(4 x + 1) x}{x (x - 2)} + \frac{2 (x - 2)}{x (x - 2)} + \frac{4}{x (x - 2)} = 0$

Step 2.5

Combine the numerators over the common denominator.

\frac{(4 x + 1) x + 2 (x - 2)}{x (x - 2)} + \frac{4}{x (x - 2)} = 0

$\frac{(4 x + 1) x + 2 (x - 2)}{x (x - 2)} + \frac{4}{x (x - 2)} = 0$

Step 2.6

Combine the numerators over the common denominator.

\frac{(4 x + 1) x + 2 (x - 2) + 4}{x (x - 2)} = 0

$\frac{(4 x + 1) x + 2 (x - 2) + 4}{x (x - 2)} = 0$

Step 2.7

Simplify each term.

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Step 2.7.1

Apply the distributive property.

\frac{4 x \cdot x + 1 x + 2 (x - 2) + 4}{x (x - 2)} = 0

$\frac{4 x \cdot x + 1 x + 2 (x - 2) + 4}{x (x - 2)} = 0$

Step 2.7.2

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 2.7.2.1

Move

x

$x$ .

\frac{4 (x \cdot x) + 1 x + 2 (x - 2) + 4}{x (x - 2)} = 0

$\frac{4 (x \cdot x) + 1 x + 2 (x - 2) + 4}{x (x - 2)} = 0$

Step 2.7.2.2

Multiply

x

$x$ by

x

$x$ .

\frac{4 x^{2} + 1 x + 2 (x - 2) + 4}{x (x - 2)} = 0

$\frac{4 x^{2} + 1 x + 2 (x - 2) + 4}{x (x - 2)} = 0$

\frac{4 x^{2} + 1 x + 2 (x - 2) + 4}{x (x - 2)} = 0

$\frac{4 x^{2} + 1 x + 2 (x - 2) + 4}{x (x - 2)} = 0$

Step 2.7.3

Multiply

x

$x$ by

1

$1$ .

\frac{4 x^{2} + x + 2 (x - 2) + 4}{x (x - 2)} = 0

$\frac{4 x^{2} + x + 2 (x - 2) + 4}{x (x - 2)} = 0$

Step 2.7.4

Apply the distributive property.

\frac{4 x^{2} + x + 2 x + 2 \cdot - 2 + 4}{x (x - 2)} = 0

$\frac{4 x^{2} + x + 2 x + 2 \cdot - 2 + 4}{x (x - 2)} = 0$

Step 2.7.5

Multiply

2

$2$ by

- 2

$- 2$ .

\frac{4 x^{2} + x + 2 x - 4 + 4}{x (x - 2)} = 0

$\frac{4 x^{2} + x + 2 x - 4 + 4}{x (x - 2)} = 0$

\frac{4 x^{2} + x + 2 x - 4 + 4}{x (x - 2)} = 0

$\frac{4 x^{2} + x + 2 x - 4 + 4}{x (x - 2)} = 0$

Step 2.8

Combine the opposite terms in

4 x^{2} + x + 2 x - 4 + 4

$4 x^{2} + x + 2 x - 4 + 4$ .

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Step 2.8.1

Add

- 4

$- 4$ and

4

$4$ .

\frac{4 x^{2} + x + 2 x + 0}{x (x - 2)} = 0

$\frac{4 x^{2} + x + 2 x + 0}{x (x - 2)} = 0$

Step 2.8.2

Add

4 x^{2} + x + 2 x

$4 x^{2} + x + 2 x$ and

0

$0$ .

\frac{4 x^{2} + x + 2 x}{x (x - 2)} = 0

$\frac{4 x^{2} + x + 2 x}{x (x - 2)} = 0$

\frac{4 x^{2} + x + 2 x}{x (x - 2)} = 0

$\frac{4 x^{2} + x + 2 x}{x (x - 2)} = 0$

Step 2.9

Add

x

$x$ and

2 x

$2 x$ .

\frac{4 x^{2} + 3 x}{x (x - 2)} = 0

$\frac{4 x^{2} + 3 x}{x (x - 2)} = 0$

Step 2.10

Factor

x

$x$ out of

4 x^{2} + 3 x

$4 x^{2} + 3 x$ .

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Step 2.10.1

Factor

x

$x$ out of

4 x^{2}

$4 x^{2}$ .

\frac{x (4 x) + 3 x}{x (x - 2)} = 0

$\frac{x (4 x) + 3 x}{x (x - 2)} = 0$

Step 2.10.2

Factor

x

$x$ out of

3 x

$3 x$ .

\frac{x (4 x) + x \cdot 3}{x (x - 2)} = 0

$\frac{x (4 x) + x \cdot 3}{x (x - 2)} = 0$

Step 2.10.3

Factor

x

$x$ out of

x (4 x) + x \cdot 3

$x (4 x) + x \cdot 3$ .

\frac{x (4 x + 3)}{x (x - 2)} = 0

$\frac{x (4 x + 3)}{x (x - 2)} = 0$

\frac{x (4 x + 3)}{x (x - 2)} = 0

$\frac{x (4 x + 3)}{x (x - 2)} = 0$

Step 2.11

Cancel the common factor of

x

$x$ .

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Step 2.11.1

Cancel the common factor.

$\frac{x (4 x + 3)}{x (x - 2)} = 0$

Step 2.11.2

Rewrite the expression.

$\frac{4 x + 3}{x - 2} = 0$

Step 3

Set the numerator equal to zero.

$4 x + 3 = 0$

Step 4

Solve the equation for

$x$ .

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Step 4.1

Subtract

$3$ from both sides of the equation.

$4 x = - 3$

Step 4.2

Divide each term in

$4 x = - 3$ by

$4$ and simplify.

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Step 4.2.1

Divide each term in

$4 x = - 3$ by

$4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of

$4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 4.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{- 3}{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{4}$